Lemma 3  [1]}
Let \(\tilde{y}^k\)  be the vector output by Algorithm  at step \(k\) , and \(y^k\)  be the optimal solution of the subproblem REF , then under Assumption  REF
\(\frac{\beta }{2}\Vert  \tilde{y}^k - y^k \Vert ^2\le H_k(\tilde{y}^k) - H_k(y^k)\le \frac{2\beta D_{\mathcal {X}}^2 (1+\delta )}{t_k+2}\\\) 
